Question

Answer Problems 1-10 without referring back to the text. Fill in the blank or answer true or false.

If the set consisting of two functions f₁ and f₂ is linearly independent on an interval I, then the Wronskian W (f₁, f₂)¹0 for all x in I.

Short answer

False

Step-by-step solution

Definition

A set of vectors is linearly independent if no vector can be expressed as a linear combination of the others.

Example  

Assume two independent functions f₁ = x and f₂= x on the interval ¡

Find the Wronskian of the functions given as,

$$\begin{gathered} W(f_1,f_2)=\left|\begin{array}{cc}{f}_1& f_2\\ {f_1}^{\varphi }& {f_2}^{\varphi }\end{array}\right| \\ W\left(x,x^2\right)=\left|\begin{array}{cc}x& x^2\\ 1& 2x\end{array}\right| \\ =x\left(2x\right)-x^2 \\ =2x^2-x^2=x^2 \end{gathered}$$

That is, $W\left(x,x^2\right)=x^2$

Note that the Wronskian

$W(x,x^2)=0forx=0$

This is a contradiction to the statement that the Wronskian $W(f_1,f_2)$for all x in / as here for one point x = 0 this statement is false.

Hence, the statement is False